This may be an obvious question, but I'm trying to understand the point of the bleed - is it to avoid having white borders? Why not just stretch or crop the image to the very edge of the document rather than allowing it to bleed over?

The purpose of the bleed-over is to allow for the fact that no cutter or cutting die is going to hit the mark perfectly every time. By having that margin of print outside of the expected cutting area, you will not wind up with a sliver of white showing when the cutter is off by a bit.